\[ \int \frac {x \left (a+b \log \left (c x^n\right )\right )}{\left (d+e x^2\right )^3} \, dx \]
Optimal antiderivative \[ \frac {b n}{8 d e \left (e \,x^{2}+d \right )}+\frac {b n \ln \left (x \right )}{4 d^{2} e}+\frac {-a -b \ln \left (c \,x^{n}\right )}{4 e \left (e \,x^{2}+d \right )^{2}}-\frac {b n \ln \left (e \,x^{2}+d \right )}{8 d^{2} e} \]
command
integrate(x*(a+b*ln(c*x**n))/(e*x**2+d)**3,x)
Sympy 1.10.1 under Python 3.10.4 output
\[ \begin {cases} \tilde {\infty } \left (- \frac {a}{4 x^{4}} - \frac {b n}{16 x^{4}} - \frac {b \log {\left (c x^{n} \right )}}{4 x^{4}}\right ) & \text {for}\: d = 0 \wedge e = 0 \\\frac {- \frac {a}{4 x^{4}} - \frac {b n}{16 x^{4}} - \frac {b \log {\left (c x^{n} \right )}}{4 x^{4}}}{e^{3}} & \text {for}\: d = 0 \\\frac {\frac {a x^{2}}{2} - \frac {b n x^{2}}{4} + \frac {b x^{2} \log {\left (c x^{n} \right )}}{2}}{d^{3}} & \text {for}\: e = 0 \\- \frac {2 a d^{2}}{8 d^{4} e + 16 d^{3} e^{2} x^{2} + 8 d^{2} e^{3} x^{4}} - \frac {b d^{2} n \log {\left (x - \sqrt {- \frac {d}{e}} \right )}}{8 d^{4} e + 16 d^{3} e^{2} x^{2} + 8 d^{2} e^{3} x^{4}} - \frac {b d^{2} n \log {\left (x + \sqrt {- \frac {d}{e}} \right )}}{8 d^{4} e + 16 d^{3} e^{2} x^{2} + 8 d^{2} e^{3} x^{4}} + \frac {b d^{2} n}{8 d^{4} e + 16 d^{3} e^{2} x^{2} + 8 d^{2} e^{3} x^{4}} - \frac {2 b d e n x^{2} \log {\left (x - \sqrt {- \frac {d}{e}} \right )}}{8 d^{4} e + 16 d^{3} e^{2} x^{2} + 8 d^{2} e^{3} x^{4}} - \frac {2 b d e n x^{2} \log {\left (x + \sqrt {- \frac {d}{e}} \right )}}{8 d^{4} e + 16 d^{3} e^{2} x^{2} + 8 d^{2} e^{3} x^{4}} + \frac {b d e n x^{2}}{8 d^{4} e + 16 d^{3} e^{2} x^{2} + 8 d^{2} e^{3} x^{4}} + \frac {4 b d e x^{2} \log {\left (c x^{n} \right )}}{8 d^{4} e + 16 d^{3} e^{2} x^{2} + 8 d^{2} e^{3} x^{4}} - \frac {b e^{2} n x^{4} \log {\left (x - \sqrt {- \frac {d}{e}} \right )}}{8 d^{4} e + 16 d^{3} e^{2} x^{2} + 8 d^{2} e^{3} x^{4}} - \frac {b e^{2} n x^{4} \log {\left (x + \sqrt {- \frac {d}{e}} \right )}}{8 d^{4} e + 16 d^{3} e^{2} x^{2} + 8 d^{2} e^{3} x^{4}} + \frac {2 b e^{2} x^{4} \log {\left (c x^{n} \right )}}{8 d^{4} e + 16 d^{3} e^{2} x^{2} + 8 d^{2} e^{3} x^{4}} & \text {otherwise} \end {cases} \]
Sympy 1.8 under Python 3.8.8 output
\[ \text {Timed out} \]________________________________________________________________________________________